Optimal switching of switched systems with time delay in discrete time

Abstract

This paper addresses a kind of optimal switching problem to minimize a quadratic cost functional for the discrete-time switched linear system with time delay. Since the dynamics is influenced by the switching sequence and the time delay, most existing gradient-based methods and relaxation techniques cannot be applied. In order to find the optimal solution, we first formulate the switched time-delay system into an equivalent switched system to separate the cross term of coefficient matrices. Based on the positive semi-definiteness of the system, we derive a series of lower bounds of the cost functional. By comparing them with the current optimal value, a depth-first branch and bound technique is proposed and the global optimal solution can be exactly obtained. Some numerical examples are demonstrated to verify the high efficiency of the method.